# Arbitrary constants in solutions of differential equations

While introducing myself to differential equations, I read that the solution to a differential equation may contain an arbitrary constant without being a general solution.

I have been solving initial value problems under the concept of anti-differentiation for a long time now. I am a little aware of general and particular solutions from my engineering classes. I have always thought that a solution having an arbitrary constant cannot be anything other than a general solution. I don't think a particular solution can have an arbitrary constant in it. The lines that I read today left me curious. I am really curious to see these equations and maybe know how they work.

• If you face a differential equation of order $n$, the general solution contains $n$ constants. Now, if you are given $m$ conditions $(m \leq n)$, $m$ of these constants "disappear" since fixed by the conditions. – Claude Leibovici Sep 10 '17 at 6:55
• So, the solution, having some constants, is now a particular solution under a few conditions( m ), right? – R004 Sep 10 '17 at 6:58
• I think what your text is trying to say is that a solution containing the right number of arbitrary constants, although it is a "general" solution (not a particular solution), yet it may fail to be the most general solution; i.e., there may be particular solutions which can not be obtained by assigning values to the arbitrary constants. – bof Sep 10 '17 at 7:01
• For example, the (non-linear) first order equation $$\frac{dy}{dt}=y^2$$ has the "general solution" $$y=\frac1{C-t}$$ but it also has the "singular solution" $$y=0$$ which is not covered by the "general solution". – bof Sep 10 '17 at 7:05
• This is a very interesting example. I appreciate it. It does justify your "most general solution" statement. But there still is a room for doubt about whether the text is exactly conveying what you just did. – R004 Sep 10 '17 at 7:13

One can find examples (not exclusively) when the solution includes multi-valuated functions, which is the sometimes the case of inverse functions. For example, the ODE : $$\cos(y(x))\frac{dy}{dx}=1$$ has this family of solutions : $$y(x)=\sin^{-1}(x+c_1)$$ This solution includes an arbitrary constant $c_1$ but is not the general solution which is : $$y(x)=\sin^{-1}(x+c_1)+2\pi\:n$$